In mathematics, the directional derivative of a multivariate differentiable function along a given vectorv at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity specified by v. It therefore generalizes the notion of a partial derivative, in which the rate of change is taken along one of the curvilinearcoordinate curves, all other coordinates being constant.

A contour plot of f(x,y)=x2+y2{\displaystyle f(x,y)=x^{2}+y^{2}}, showing the gradient vector in green, and the unit vector u{\displaystyle {\mathbf {u}}} scaled by the directional derivative in the direction of u{\displaystyle {\mathbf {u}}} in orange. The gradient vector is longer because the gradient points in the direction of greatest rate of increase of a function.

where the ∇{\displaystyle \nabla } on the right denotes the gradient and ⋅{\displaystyle \cdot } is the dot product.[3] Intuitively, the directional derivative of f at a point x represents the rate of change of f, in the direction of v with respect to time, when moving past x.

The angle α between the tangent A and the horizontal will be maximum if the cutting plane contains the direction of the gradient A.

In a Euclidean space, some authors[4] define the directional derivative to be with respect to an arbitrary nonzero vector v after normalization, thus being independent of its magnitude and depending only on its direction.[5]

This definition gives the rate of increase of f per unit of distance moved in the direction given by v. In this case, one has

The Lie derivative of a vector field Wμ(x){\displaystyle \scriptstyle W^{\mu }(x)} along a vector field Vμ(x){\displaystyle \scriptstyle V^{\mu }(x)} is given by the difference of two directional derivatives (with vanishing torsion):

Directional derivatives are often used in introductory derivations of the Riemann curvature tensor. Consider a curved rectangle with an infinitesimal vector δ along one edge and δ′ along the other. We translate a covector S along δ then δ′ and then subtract the translation along δ′ and then δ. Instead of building the directional derivative using partial derivatives, we use the covariant derivative. The translation operator for δ is thus

Here ϵ⋅∇{\displaystyle {\boldsymbol {\epsilon }}\cdot \nabla } is the directional derivative along the infinitesimal displacement ε. We have found the infinitesimal version of the translation operator:

As a technical note, this procedure is only possible because the translation group forms an Abeliansubgroup (Cartan subalgebra) in the Poincaré algebra. In particular, the group multiplication law U(a)U(b)=U(a+b) should not be taken for granted. We also note that Poincaré is a connected Lie group. It is a group of transformations T(ξ) that are described by a continuous set of real parameters ξa{\displaystyle \scriptstyle \xi ^{a}}. The group multiplication law takes the form

The actual operators on the Hilbert space are represented by unitary operators U(T(ξ)). In the above notation we suppressed the T; we now write U(λ) as U(P(λ)). For a small neighborhood around the identity, the power series representation

The rotation operator also contains a directional derivative. The rotation operator for an angle θ, i.e. by an amount θ=|θ| about an axis parallel to θ^{\displaystyle \scriptstyle {\hat {\theta }}}=θ/θ is

A normal derivative is a directional derivative taken in the direction normal (that is, orthogonal) to some surface in space, or more generally along a normal vector field orthogonal to some hypersurface. See for example Neumann boundary condition. If the normal direction is denoted by n{\displaystyle {\mathbf {n}}}, then the directional derivative of a function f is sometimes denoted as ∂f∂n{\displaystyle {\frac {\partial f}{\partial n}}}. In other notations

Several important results in continuum mechanics require the derivatives of vectors with respect to vectors and of tensors with respect to vectors and tensors.[13] The directional directive provides a systematic way of finding these derivatives.

The definitions of directional derivatives for various situations are given below. It is assumed that the functions are sufficiently smooth that derivatives can be taken.

Let f(S){\displaystyle f({\boldsymbol {S}})} be a real valued function of the second order tensor S{\displaystyle {\boldsymbol {S}}}. Then the derivative of f(S){\displaystyle f({\boldsymbol {S}})} with respect to S{\displaystyle {\boldsymbol {S}}} (or at S{\displaystyle {\boldsymbol {S}}}) in the direction T{\displaystyle {\boldsymbol {T}}} is the second order tensor defined as

Let F(S){\displaystyle {\boldsymbol {F}}({\boldsymbol {S}})} be a second order tensor valued function of the second order tensor S{\displaystyle {\boldsymbol {S}}}. Then the derivative of F(S){\displaystyle {\boldsymbol {F}}({\boldsymbol {S}})} with respect to S{\displaystyle {\boldsymbol {S}}} (or at S{\displaystyle {\boldsymbol {S}}}) in the direction T{\displaystyle {\boldsymbol {T}}} is the fourth order tensor defined as

^If the dot product is undefined, the gradient is also undefined; however, for differentiable f, the directional derivative is still defined, and a similar relation exists with the exterior derivative.